Study of entanglement via a multi-agent dynamical quantum game

At both conceptual and applied levels, quantum physics provides new opportunities as well as fundamental limitations. We hypothetically ask whether quantum games inspired by population dynamics can benefit from unique features of quantum mechanics such as entanglement and nonlocality. For doing so, we extend quantum game theory and demonstrate that in certain models inspired by ecological systems where several predators feed on the same prey, the strength of quantum entanglement between the various species has a profound effect on the asymptotic behavior of the system. For example, if there are sufficiently many predator species who are all equally correlated with their prey, they are all driven to extinction. Our results are derived in two ways: by analyzing the asymptotic dynamics of the system, and also by modeling the system as a quantum correlation network. The latter approach enables us to apply various tools from classical network theory in the above quantum scenarios. Several generalizations and applications are discussed.

and where it is assumed, without loss of generality, that η is real. In what follows it is assumed that ρ is non-negative. As C is a correlation matrix it must be positive semidefinite. The block decomposition of C allows for an equivalent requirement, M (n, p, ρ) 0, D − Γ T M (n, p, ρ) −1 Γ 0 (4) known as Schur complement condition for positive semidefiniteness. As C is a random matrix there should be some distribution of the largest possible ρ admitting the conditions (4). A known result in random matrix theory, based on concentration of measure arguments, has it that the distribution of the eigenvalues of A(n, p)/ √ np approaches the Wigner semicircle distribution with radius 2 in the n → ∞ limit (see Theorem 2.8.1 in [1], and Exercise 3.1 in [2]), Therefore, for sufficiently large n it follows that where γ ∼ Wsc(γ). Furthermore, that M (n, p, ρ) 0 means ρ < −1/(γ √ np) = O(1/ √ np), for γ < 0 and sufficiently large n. The additional condition in (4) further restricts the bound on ρ. To find out how we shall express the distribution of ρ * , the maximum value of ρ, such that (4) hold.
From (4) and (5) it follows that for large n, where δ is a discrete random variable satisfying δ ≤ (n − np )/ np (1 − p)/p. The expression in the numerator on the right follows from known concentration inequalities concerning sums of Bernoulli random variables; Thus, for large n, the sum The random variable δ accounts for the uncertainty in these estimates.
The right hand side in (6) is equivalent to non-negativity of the trace and determinant of the underlying matrix difference. This leads to, where g(η, p) the same one defined in the theorem. In other words, We thus may identify the right hand side above as the random variable ρ * √ np. Therefore, which holds for sufficiently large n.
Because γ is distributed according to Wsc(γ) in the n → ∞ limit, and similarly |γ| is distributed according to Wsc + (|γ|), it follows from (10) that Note that by the very definitions of ζ and g(η, p), the product ζg(η, p) = O( √ 1 − η). Finally, in the n → ∞ limit, from which the theorem follows.

Nonlocality and dynamics
Linearization Locally, the system may be described by its Lyapunov exponents, the eigenvalues of exp(Jt) with the Jacobian J, where Let us assume that ∀i ∈ [n] , ζ i = ζ, v i = v; and also ∀i ∈ {2, . . . , n} , B i = B. Note that B 1 may be different than the others! Rewriting the Jacobian: The characteristic polynomial of J: Which implies that λ mul = Bc − ζ appears as an eigenvalue with algebraic multiplicity of (at least) n − 2. λ mul corresponds to dynamical modes for which c, v 1 and n k=2 v k are constant; i.e., the only dynamics are amongst the n − 1 "homogeneous" virus species, and therefore not particularly interesting for our purposes.
The other three eigenvalues are the solutions of the cubic equation: We shall examine the dynamics of the system near a point where the normalized populations are all 1. Moreover, we are only interested in cases where the viruses all have positive population payoffs, i.e. B 1 , B > 0. First, let us rewrite (17) using the variableλ := λ + ζ: Now, we substitute sin θ, c = v = 1, and also define δ := γ + ζ: Generally, given parameters β, γ, ζ, θ, one may find some value n c = n(β, γ, ζ, θ), such that for any n ≤ n c the system would admit at least one non-negative Lyapunov exponent (note for some range of the parameters we would have n c = 0). This function n(β, γ, ζ, θ) is given implicitly by considering λ max = ζ, whereλ max is the solution to (19) having the largest real part.

Non-entangled case
In the non-entangled (equally-correlated) case, we assume B CV k = 2 2/n =: B for all k, implying B 1 = B = β/ √ 2n. We may define a new dynamic variable:v ( n i=1 v i ) /n, and replace our system of n + 1 equations with only two:ċ This system admits the following equilibrium point: and the following Jacobian: Let us find its eigenvalues. The characteristic polynomial is: Let us use the notation δ = γ + ζ and assume c ≈ 1, v tot ≈ 1: The discriminant is: which vanishes for δ ± = β(n+1) √ 2n ± √ 2β. For large enough values of n we have: Thus, it follows that λ + ≈ −ζ + δ 2 2 √ 2nβ and λ − ≈ γ − β n 2 + 2β 2 −δ 2 2 √ 2nβ ≈ γ − 2β/B. Now we shall find the eigenvectors: Thus we have: In the maximally-entangled case, B CV1 = 2 √ 2 and for any k > 1, B CV k = 0. Thus, B 1 = β/ √ 2 and B = 0. The system admits the following equilibrium point: The eigenvalues are λ mul = −ζ with algebraic multiplicity n − 2, and the other three are solutions of the following cubic equation: So we see that λ = −ζ actually has algebraic multiplicity n − 1, and the remaining two eigenvectors are the roots of: Let us assume that v ≈ c ≈ 1 . Using these approximations, the Jacobian has the form: and the remaining two eigenvalues are: where we have used the notation δ = γ + ζ. The corresponding eigenvectors are: And for λ mul we have the eigenvectors {e j } n j=2 , where e j is the vector with jth entry 1 and all other entries 0 (note the indices start from 0).
1. γ + ζ = 4B 1 = 2 √ 2β: we obtain the eigenvalue λ 2 = (γ − ζ) /2 with algebraic multiplicity 2. Let us express it with β rather than ζ: while the eigenvalue λ mul , with multiplicity n − 1, obtains the form: Compare with the Lyapunov exponents of the non-entangled case: 2β. Interestingly, for n = 4 the cells have the same Lyapunov exponents in the non-entangled and maximally-entangled states, up to terms in the order of 1/ √ n.